In this work we present a pseudo-random Bit Generator via unidimensional multi-modal discrete dynamical systems called k-modal maps. These multimodal maps are based on the logistic map and are useful to yield pseudo-random sequences with longer period, i.e., in order to attend the problem of periodicity. In addition the pseudo-random sequences generated via multi-modal maps are evaluated with the statistical suite of test from NIST and satisfactory results are obtained when they are used as key stream. Furthermore, we show the impact of using these sequences in a stream cipher resulting in a better encryption quality correlated with the number of modals of the chaotic map. Finally, a statistical security analysis applied to cipher images is given. The proposed algorithm to encrypt is able to resist the chosen-plaintext attack and differential attack because the same set of encryption keys generates a different cipher image every time it is used.

Abstract. Given a graded complete intersection ideal J = (f 1 , . . . , fc) ⊆ k[x 0 , . . . , xn] = S, where k is a field of characteristic p > 0 such that [k : k p ] < ∞, we show that if S/J has an isolated non-F-pure point then the Frobenius action on top local cohomology H n+1−c m (S/J) is injective in sufficiently negative degrees, and we compute the least degree of any kernel element. If S/J has an isolated singularity, we are also able to give an effective bound on p ensuring the Frobenius action on H n+1−c m (S/J) is injective in all negative degrees, extending a result of Bhatt and Singh in the hypersurface case.

We provide a family of examples where the F -pure threshold and the log canonical threshold of a polynomial are different, but where p does not divide the denominator of the F -pure threshold (compare with an example of Mustaţȃ-Takagi-Watanabe). We then study the F -signature function in the case where either the F -pure threshold and log canonical threshold coincide or where p does not divide the denominator of the F -pure threshold. We show that the F -signature function behaves similarly in those two cases. Finally, we include an appendix which shows that the test ideal can still behave in surprising ways even when the F -pure threshold and log canonical threshold coincide.

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